Theorem cofu2nd 16941

Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuval.b	⊢ 𝐵 = (Base‘𝐶)
cofuval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
cofu2nd	⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))

Proof of Theorem cofu2nd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofuval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		cofuval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		cofuval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4	1, 2, 3	cofuval 16938	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
5	4	fveq2d 6452	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩))
6		fvex 6461	. . . . 5 ⊢ (1st ‘𝐺) ∈ V
7		fvex 6461	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
8	6, 7	coex 7399	. . . 4 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V
9	1	fvexi 6462	. . . . 5 ⊢ 𝐵 ∈ V
10	9, 9	mpt2ex 7529	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V
11	8, 10	op2nd 7456	. . 3 ⊢ (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
12	5, 11	syl6eq 2830	. 2 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))))
13		simprl 761	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
14	13	fveq2d 6452	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋))
15		simprr 763	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
16	15	fveq2d 6452	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((1st ‘𝐹)‘𝑦) = ((1st ‘𝐹)‘𝑌))
17	14, 16	oveq12d 6942	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) = (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)))
18	13, 15	oveq12d 6942	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥(2nd ‘𝐹)𝑦) = (𝑋(2nd ‘𝐹)𝑌))
19	17, 18	coeq12d 5534	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
20		cofu2nd.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
21		cofu2nd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
22		ovex 6956	. . . 4 ⊢ (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∈ V
23		ovex 6956	. . . 4 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
24	22, 23	coex 7399	. . 3 ⊢ ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)) ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)) ∈ V)
26	12, 19, 20, 21, 25	ovmpt2d 7067	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404   ∘ ccom 5361  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  1^st c1st 7445  2^nd c2nd 7446  Basecbs 16266   Func cfunc 16910   ∘_func ccofu 16912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-map 8144  df-ixp 8197  df-func 16914  df-cofu 16916

This theorem is referenced by:  cofu2  16942  cofucl  16944  cofuass  16945  cofull  16990  cofth  16991  catciso  17153